Functional Dependence and the Method of Alternate Formulations in Optimal Design

1992 ◽  
Vol 114 (4) ◽  
pp. 596-602
Author(s):  
C. R. Hammond ◽  
G. E. Johnson
Keyword(s):  
Optimal Design ◽  
Functional Dependence ◽  
Problem Formulation ◽  
System Model ◽  
Decision Variable ◽  
Reduced Gradient Method ◽  
State Variable ◽  
Mathematical Properties ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

In an earlier article in this journal we introduced the Method of Alternate Formulations (MAF). MAF is a nonnumerical approach to constrained optimal design implemented with symbolic mathematics. The MAF problem formulation is the same as is used by the generalized reduced gradient method. There are usually many ways to partition the design vector into decision variable and state variable components and so there are usually many different alternate formulations for the objective function and constraints. Each alternate formulation contains all of the information about the physical system. Yet all other mathematical properties (e.g., convexity, linearity, scaling, etc.) can change. It has been observed that some of the alternate formulations that should exist based strictly on the theory of combinations cannot be obtained. In this paper, we show that this phenomenon occurs whenever there is functional dependence in the system model. Several examples are used to show how functional dependence affects the search for the solution by MAF. Prediction of functional dependence at the outset informs the designer which formulations cannot exist. This allows the designer to concentrate effort (more productively) on other formulations of the problem.

The Method of Alternate Formulations Applied to Optimization Problems With Functionally Dependent State Equations

1989 ◽  
Author(s):  
C. R. Hammond ◽  
G. E. Johnson
Keyword(s):  
Optimization Problems ◽  
Functional Dependence ◽  
Numerical Approach ◽  
Upper And Lower Bounds ◽  
Decision Variable ◽  
State Equations ◽  
Reduced Gradient Method ◽  
Mathematical Properties ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

Abstract The Method of Alternate Formulations (MAF) is a non-numerical approach to constrained optimal design. MAF requires that the problem statement be transformed into an objective function, a set of equality-constraints (i.e. state equations), and a set of upper and lower bounds on the variables. In this format, the design vector can be partitioned into decision variable and state variable components. This is the same format as used in the solution of such problems by the generalized reduced gradient method. The fact that there are usually several ways to effect the partition of the design vector gives rise to the existence of alternate formulations. Each alternate formulation contains all of the information about the physical system — and the constrained optima are invariant under the transformation from form to form. Yet all other mathematical properties (e.g., convexity, linearity, scaling, etc.) can change. In this paper, we consider the special case when the state equations are functionally dependent, hence some of the expected constraint intersections do not exist. Several examples are used to demonstrate the concept of functional dependence and to show how functional dependence affects the search for the solution.

The Generalized Reduced Gradient Method: A Reliable Tool for Optimal Design

1977 ◽  
Vol 99 (2) ◽  
pp. 394-400 ◽  
Author(s):  
G. A. Gabriele ◽  
K. M. Ragsdell
Keyword(s):  
Optimal Design ◽  
Gradient Method ◽  
Theoretical Development ◽  
Test Problems ◽  
Basic Algorithm ◽  
Reduced Gradient Method ◽  
Fortran Code ◽  
Reduced Gradient ◽  
Efficient Code ◽  
Generalized Reduced Gradient

This paper is a presentation of a method, called the Generalized Reduced Gradient Method, which has not received wide attention in the engineering design literature. Included is a theoretical development of the method, a description of the basic algorithm, and additional recommendations to produce an efficient code. A Fortran code employing this theory was written and tested on the Eason and Fenton [1] test problems, illustrating the method to be efficient and reliable.

Optimal Design of Mechanical Components With Sizing, Configurational and Topological Consideration

1989 ◽  
Author(s):  
E. Sandgren ◽  
H. Lee ◽  
M. El-Sayed
Keyword(s):  
Optimal Design ◽  
Frame Structure ◽  
Optimization Approach ◽  
Reduced Gradient Method ◽  
Beam Elements ◽  
Reduced Gradient ◽  
Mechanical Components ◽  
Upper Level ◽  
Generalized Reduced Gradient ◽  
Seat Frame

Abstract A methodology for the optimal design of structural mechanical components with sizing, configurational and topological consideration is presented. The optimization involves the minimization of structural weight subject to stress and deflection constraints. A finite element code with internally generated design sensitivities and general element connectivities is implemented in order to solve for the stresses and deflections in the structure. Beam elements with various crossectional configurations are considered. The generalized reduced gradient method is applied to solve the constrained nonlinear programming problem. A hierarchical design optimization approach is taken with topological issues considered at the upper level and geometric and crossectional issues considered at the lower level. An example involving the design of an automotive seat frame structure is presented to demonstrate the approach.

Optimal Design of a Class of Welded Structures Based on Design for Latitude

1985 ◽  
Vol 107 (4) ◽  
pp. 482-487 ◽  
Author(s):  
E. Sandgren ◽  
G. Gim ◽  
K. M. Ragsdell
Keyword(s):  
Optimal Design ◽  
Objective Function ◽  
Gradient Method ◽  
Low Cost ◽  
Optimal Solution ◽  
Beam Structures ◽  
Welded Structures ◽  
Reduced Gradient Method ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

The minimization of the sensitivity of a design to variations in uncontrollable parameters is illustrated. The procedure is applied to the design of a class of welded beam structures which results in a low-cost design with minimal sensitivities. Dominant constraints are chosen which contain variations of the uncontrollable parameters. A dual objective function is formed and tradeoff curves are presented from which the optimal solution is selected. The minimization is carried out using the generalized reduced gradient method and other applications are presented.

Optimal Design of Semisubmersible’s Form Based on Systems Analysis

1984 ◽  
Vol 106 (4) ◽  
pp. 524-530 ◽  
Author(s):  
S. Akagi ◽  
R. Yokoyama ◽  
K. Ito
Keyword(s):  
Optimal Design ◽  
Computer Aided Design ◽  
Optimization Problem ◽  
Design Method ◽  
Gradient Algorithm ◽  
Interpretive Structural Modeling ◽  
Reduced Gradient ◽  
Optimal Design Method ◽  
Generalized Reduced Gradient ◽  
Aided Design

With the objective of developing a computer-aided design method to seek the optimal semisubmersible’s form, hierarchical relationships among many design objectives and conditions are investigated first based on the interpretive structural modeling method. Then, an optimal design method is formulated as a nonlinear multiobjective optimization problem by adopting three mutually conflicting design objectives. A set of Pareto optimal solutions is derived numerically by adopting the generalized reduced gradient algorithm, and it is ascertained that the designer can determine the optimal form more rationally by investigating the trade-off relationships among design objectives.

Synthesis of a Large Oscillation Four-Bar Mechanism

1997 ◽  
Author(s):  
Gloria K. Starns ◽  
Donald R. Flugrad
Keyword(s):  
Reduced Gradient Method ◽  
Large Oscillation ◽  
Oscillation Mechanism ◽  
Straight Line ◽  
Transmission Angle ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient ◽  
Angular Oscillations ◽  
Four Bar Linkage ◽  
Effective Transmission

Abstract This paper demonstrates procedures implemented for the synthesis of a four-bar mechanism that produces large angular oscillations of the output member while maintaining effective transmission angles. The mechanisms are modeled as being driven by a force applied at the coupler link. Additionally this force’s line of action is constrained to occur along an approximate straight line. This research was conducted out of the need for a device that is capable of retraction of the horizontal tool bar housed on the back of a tractor. The tool bars accommodate the implements required to accomplish the numerous tasks of the farmer, i.e. row markers, sprayer arms, planters, etc. Upon retraction of the tool bar so that it is parallel to ground, the appropriate tools are lowered to their working position. As the length of these bars increases, a savings of time and increased productivity is realized. Kurt Hain makes the following observation regarding large oscillation mechanisms in [1]: “It would be very difficult to solve this problem with one four-bar linkage, because it is difficult to design a four-bar linkage having such a large oscillation of a crank without running into problems of poor transmission angle characteristics; it might be possible to use linkages in combinations with gears, but this would make the mechanism more expensive, less efficient, and probably noisier.” In this study simulated annealing, a genetic algorithm and the generalized reduced gradient method are used to produce mechanisms with large angular oscillations of the output member and transmission angles that vary by as little as 20° from 90°. A comparative analysis of each of the optimization procedures is presented with observations regarding the efficacy of each method in the solution of the large oscillation mechanism.

An inventory model of a three parameter Weibull distributed deteriorating item with variable demand dependent on price and frequency of advertisement under trade credit

2019 ◽  
Vol 53 (3) ◽  
pp. 903-916 ◽  
Author(s):  
Ali Akbar Shaikh ◽  
Leopoldo Eduardo Cárdenas–Barrón ◽  
Asoke Kumar Bhunia ◽  
Sunil Tiwari
Keyword(s):  
Trade Credit ◽  
Inventory Model ◽  
Selling Price ◽  
Credit Policy ◽  
Variable Demand ◽  
Reduced Gradient Method ◽  
Deterioration Rate ◽  
Rate Dependent ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

This paper develops an inventory model for a deteriorating item with variable demand dependent on the selling price and frequency of advertisement of the item under the financial trade credit policy. Shortages are allowed and these are partially backlogged with a variable rate dependent on the duration of waiting time until to the arrival of next order. In this inventory model, the deterioration rate follows a three-parameter Weibull distribution. The corresponding inventory model is formulated and solved by using the well-known generalized reduced gradient method along with an algorithm. To validate the inventory model, two numerical examples are considered and solved. Finally, based on one numerical example, the impacts of different parameters are studied by a sensitivity analysis considering one parameter at a time and leaving the other parameters fixed.

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